INDEX OF AN IMPLICIT DIFFERENTIAL EQUATION

In this paper we introduce the concept of the index of an implicit differential equation F(x,y,p) = 0, where F is a smooth function, p = dy/dx, F(p) = 0 and F(pp) = 0 at an isolated singular point. We also apply the results to study the geometry of surfaces in R(5).

A further exploration of a nucleophilicity index based on the gas-phase ionization potentials

An empirical nucleophilicity index based on the gas-phase ionization potentials has been recently shown to be useful categorizing and settling the nucleophilicity power of a series of captodative ethylenes reacting in cycloaddition reactions (L.R. Domingo, E. Chamorro, P. Perez, Journal of Organic Chemistry 73 (2008) 4615-4624). In the present work, the applicability of such model is tested within a broader series of substituted alkenes, substituted aromatic compounds and simple nucleophilic molecules. This index obtained within a Koopman`s theorem framework has been evaluated here in both gas and solution phases for several well-known nucleophiles. These results are found to be linearly correlated. Finally, the feasibility of the predictive character of this index has been discussed in comparison to the available experimental nucleophilicities of some amines in water. These results further support and validate the usefulness of such approximation in the modeling of the global nucleophilicity. (C) 2008 Elsevier B.V. All rights reserved.

Histological analyses of thermal effect caused by 1.2 W diode laser irradiation at rat periodontal pockets

The aim of this in vivo study was to evaluate the thermal effects caused by 810 nm 1.2 W diode laser irradiation of periodontal tissues. Despite all data available concerning the laser application for periodontal treatment, one of the most relevant challenges is to prevent the harmful tissue heating induced by different clinical protocols. Periodontal pockets were induced at molars in 96 rats. Several irradiation powers under CW mode were investigated: 0, 400, 600, 800, 1000, 1200 mW. The pockets were irradiated using a 300 A mu m frontal illumination fiber. The animals were killed at 4 or 10 days after irradiation. The mandible was surgically removed and histologically processed. The histological sections stained with H/E demonstrated that irradiation parameters up to 1000 mW were thermally safe for the periodontal tissues. The sections stained with Brown & Brenn technique evidenced bacteria in the periodontal tissues. Consequently, the diode laser irradiation as a unique treatment was not capable to eliminate bacteria of the biofilm present in the pockets. According to the methodology used here, it was concluded that the thermal variation promoted by a diode laser can cause damage to periodontal tissues depending on the energy density used. The 1.2 W diode laser irradiation itself does not control the bacteria present in the biofilm of the periodontal pockets without mechanical action. The knowledge of proper high intensity laser parameters and methods of irradiation for periodontal protocols may prevent any undesirable thermal damage to the tissues.

Nonlinear refractive index of RECOB (RE = Gd and La) crystals

The Z-scan technique is employed to obtain the nonlinear refractive index (n (2)) of the Ca(4)REO(BO(3))(3) (RECOB, where RE = Gd and La) single crystals using 30 fs laser pulses centered at 780 nm for the two orthogonal orientations determined by the optical axes (X and Z) relative to the direction of propagation of the laser beam (k//Y// crystallographic b-axis). The large values of n (2) indicate that both GdCOB and LaCOB are potential hosts for Yb:RECOB lasers operating in the Kerr-lens mode locking (KLM) regime.

Time-resolved study electronic and thermal contributions to the nonlinear refractive index of Nd3+: SBN laser crystals

The propagation of an optical beam through dielectric media induces changes in the refractive index, An, which causes self-focusing or self-defocusing. In the particular case of ion-doped solids, there are thermal and non-thermal lens effects, where the latter is due to the polarizability difference, Delta alpha, between the excited and ground states, the so-called population lens (PL) effect. PL is a pure electronic contribution to the nonlinearity, while the thermal lens (TL) effect is caused by the conversion of part of the absorbed energy into heat. In time-resolved measurements such as Z-scan and TL transient experiments, it is not easy to separate these two contributions to nonlinear refractive index because they usually have similar response times. In this work, we performed time-resolved measurements using both Z-scan and mode mismatched TL in order to discriminate thermal and electronic contributions to the laser-induced refractive index change of the Nd3+-doped Strontium Barium Niobate (SrxBa1-xNb2O6) laser crystal. Combining numerical simulations with experimental results we could successfully distinguish between the two contributions to An. (C) 2007 Elsevier B.V. All rights reserved.

NIELSEN COINCIDENCE THEORY OF FIBRE-PRESERVING MAPS AND DOLD`S FIXED POINT INDEX

Let M -> B, N -> B be fibrations and f(1), f(2): M -> N be a pair of fibre-preserving maps. Using normal bordism techniques we define an invariant which is an obstruction to deforming the pair f(1), f(2) over B to a coincidence free pair of maps. In the special case where the two fibrations axe the same and one of the maps is the identity, a weak version of our omega-invariant turns out to equal Dold`s fixed point index of fibre-preserving maps. The concepts of Reidemeister classes and Nielsen coincidence classes over B are developed. As an illustration we compute e.g. the minimal number of coincidence components for all homotopy classes of maps between S(1)-bundles over S(1) as well as their Nielsen and Reidemeister numbers.

On a product formula for the Conley-Zelmder index of symplectic paths and its applications

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley-Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.

UNIT VECTOR FIELDS ON ANTIPODALLY PUNCTURED SPHERES: BIG INDEX, BIG VOLUME

We establish in this paper a lower bound for the volume of a unit vector field (v) over right arrow defined ou S(n) \ {+/-x}, n = 2,3. This lower bound is related to the sum of the absolute values of the indices of (v) over right arrow at x and -x.

Maslov index in semi-Riemannian submersions

We study focal points and Maslov index of a horizontal geodesic gamma : I -> M in the total space of a semi-Riemannian submersion pi : M -> B by determining an explicit relation with the corresponding objects along the projected geodesic pi omicron gamma : I -> B in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.

Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

Given a Lorentzian manifold (M,g), a geodesic gamma in M and a timelike Jacobi field Y along gamma, we introduce a special class of instants along gamma that we call Y-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the Y-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field Y is obtained as the restriction of a globally defined timelike Killing vector field.

COMPARISON RESULTS FOR CONJUGATE AND FOCAL POINTS IN SEMI-RIEMANNIAN GEOMETRY VIA MASLOV INDEX

We prove an estimate on the difference of Maslov indices relative to the choice of two distinct reference Lagrangians of a continuous path in the Lagrangian Grassmannian of a symplectic space. We discuss some applications to the study of conjugate and focal points along a geodesic in a semi-Riemannian manifold.